Unsteady mix convectional stagnation point flow of nanofluid over a movable electro-magnetohydrodynamics Riga plate numerical approach

The flow at a time-independent separable stagnation point on a Riga plate under thermal radiation and electro-magnetohydrodynamic settings is examined in this research. Two distinct base fluids-H2O and C2H6O2 and TiO2 nanostructures develop the nanocomposites. The flow problem incorporates the equations of motion and energy along with a unique model for viscosity and thermal conductivity. Similarity components are then used to reduce these model problem calculations. The Runge Kutta (RK-4) function yields the simulation result, which is displayed in graphical and tabular form. For both involved base fluid theories, the nanofluids flow and thermal profiles relating to the relevant aspects are computed and analyzed. According to the findings of this research, the C2H6O2 model heat exchange rate is significantly higher than the H2O model. As the volume percentage of nanoparticles rises, the velocity field degrades while the temperature distribution improves. Moreover, for greater acceleration parameters, TiO2/ C2H6O2has the highest thermal coefficient whereas TiO2/ H2O has the highest skin friction coefficient. The key observation is that C2H6O2 base nanofluid has a little higher performance than H2O nanofluid.

several nanofluids with distinct flow arrangement were also actively investigated. The blended copper-alumina/ water, titania-alumina/water, and copper-titania/water nanofluids subjected to an exponential form velocity stretching sheet were numerically and statistically studied by Hussain et al. 16 . Bhatti et al. 17 also investigated the composite diamond-silica/water nanofluid with an approach toward solar collector implementations. Tripathi et al. 18 and Zeeshan et al. 19 explored another analysis of composite nanofluid flow with microfluidic channels and lubricating oil utilization. Besides this, Hussain et al. [20][21][22][23][24][25][26] , Nasir et al. 27,28 have recently published work on nanofluids boundary layer flows in several configurations.
The revolutionary phenomena known as stagnation point flow configuration is frequently seen when substance is hit to a solid surface perpendicularly or diagonally. Diverse kinds of utilization in scientific and technical concepts, such as aircraft wings and oscillatory processes, have been supported by the stagnation point hypothesis. Additionally, the designing of automotive and numerous manufacturing operations also emphasized the significance of stagnation point streams. To correctly assess the implementations, it is crucial to understand the mathematical foundation and modeling of such phenomena. The influence of fluid motion vs various shapes is reported in many studies in the literature [29][30][31][32] . Scientists appreciate steady flow in manufacturing procedures although it renders the processes easier to handle. However, real-world experience reveals that unfavorable unsteady consequences can still happen close to a gadget even in the ideal scenario of fluid flow. These unfavorable outcomes may be brought on by self-inflicted physical movements or by fluid imbalances. The two-dimensional unsteady separable stagnation point unsteady stagnation flow regarding the non-porous medium was first proposed by Ma and Hui 33 . In addition, Berrouk et al. 34 and Zainal et al. 35 performed multiple computational analyses of unsteady stagnation flow on the conventional and nanofluids.
Among the most recent advancements for addressing poor fluid conductance was the Riga plate; for further information, see Gailitis and Lielausus 36 . This gadget uses magnets and alternated sets of electrodes as an electromagnetic actuator. To control liquid motion, it is utilized to create an electromotive force which originates in the Lorentz force. Additionally, the Riga plate can be utilized to reduce surface friction and stop the development of instability 37 . According to a survey of the literature, Perez et al. 38 have already experimented with the flow of nanofluid across a Riga plate. Using a Riga plate, Supian et al. 39 investigate the electromagnetic slipping viscous flow of nanofluid. Researchers have also explored nanofluids to examine its features when the flow is structured over a Riga plate due to the problems regarding the progression of heat transmission. The radiative Hiemenz flow of copper-alumina/water nanofluid toward an EMHD Riga plate was examined by Bilal et al. 40 . The respective studies have contributed to some of the most recent research: Ragupathi et al. [41][42][43] and Hakeem et al. 44,45 .
Following a discussion of the uses of stagnation point flows, Riga plates and nanofluids, a description on contemporary investigation examines the effects of various thermal properties on the unsteady nanofluid flow that develops a stagnation point flow over a Riga plate. Aspect of the present investigation that can be estimated as follows: • The Riga plate-subjected nanofluid stagnation point analysis is seen.
• By analyzing implementations of the heat radiation and dissipation phenomenon, the thermal influence of nanofluids is observed. • Two distinct base fluids-H 2 O and C 2 H 6 O 2 and Al 2 O 3 nanostructures develop the nanocomposites. • A unique model for viscosity and thermal conductivity is utilized.
• The unsteady flow condition is also taken into consideration.
• The simulations for the nonlinear model are computed using the two numerical methods that have shown the most promising, RK-4 and CVFEM schemes.
It has been noted that various researchers have investigated that various nanofluids affect stagnation points flow. The applications of dissipation, thermal radiative phenomena, and unsteady effects for nanofluid (H 2 O/ TiO 2 and C 2 H 6 O 2 /TiO 2 ) stagnation point flow have not yet been explored. These objectives are the foundation of this investigation. Through various graphs and tables, the physical impact of flow parameters is taken into consideration. The study suggests implications in recently developed magnetically nano biosensors 38 and molecular tracking monitoring processes that make use of Riga sensing and magnetic nanocomposites in biochemical manufacturing 39 .

Physical and mathematical formulation
Time dependent, natural convection, incompressible and stagnation state flow of nanofluids H 2 O/TiO 2 and C 2 H 6 O 2 /TiO 2 is presented in Fig. 1a are examined subjected to a movable plate under the following some basic assumptions 29,34 : Here u 0 (t) = ∂x(t)/∂t (velocity of moving plate),t, t rf , γ , are time, reference time, acceleration parameters, unsteadiness parameters with = 0 (steady case), > 0(unsteadiness accelerating case), < 0 > 0(unsteadiness decelerating case) 29,34 . • In present mathematical model the electro-magnetohydrodynamic effect from the Riga plate is denoted by • T w , T ∞ are the wall and free space temperatures.
• Energy expression involves the phenomena of thermal radiation and dissipation.
The preceding formulations are used to model the current problem 29 www.nature.com/scientificreports/ Here u and v are velocities of nanofluids, g(gravitational acceleration) j 0 (current density), y 1 (electrodes), q r is radiative flux. The subscribed nf implies simple nanofluid. The Roseland approximations yield the following formulas for radiative flux 27 : where σ * signify (Stefan-Boltzman constant) and k * indicate (mean absorption coefficient). Table 1 displays the relationships of the experimentally verified hybrid nanofluid features. A list of the characteristics for H 2 O, C 2 H 6 O 2 , and TiO 2 for the numerical simulations.
Here for µ nf &k nf we introduce models 12 for base fluids C 2 H 6 O 2 and H 2 O.  The model equations and boundary conditions of current problem nondimensional version is: In the above equations the prime represents differentiation with respect to η . The obtained dimensionless parameters are presented as: where,

Solution methodology
Control volume finite element method (CVFEM) procedure. The Finite element approach, based on the control volume algorithm, is used to computationally carry out the entire simulation, together with the non-dimensional system of equations and their boundary conditions. A recently created method called CVFEM aims to get at a realistic numerical solution to the non-linear system of partial differential equations (PDEs).  Figure 1a shows the geometry of the problem and Fig. 1b is the Grids representation obtained from the CVFEM technique. A schematic diagram illustrating the CVFEM methodology is presented in Fig. 2.

RK (Runge-Kutta) procedure.
It is clearly obvious from the fact that transformed differential Eqs. (10), (11) and (12) are extremely non-linear, and it is almost tough to calculate their exact solutions in most situations.
To get the approximated solutions to these problems, we generally use a numerical method called the shooting technique with the aid of the fourth order R-K (Runge-Kutta) method 49,50 . By declaring a new set of dependent variables, we convert the existing Eqs. (10), (11) and (12) into a set of first order ODEs, as 49 :  Fig. 3a,b. The Riga concept's magnets and electrodes are organized in such a way that the resultant Lorentz forces drive the motion of the investigated electrically conducting nanofluid, allowing this analysis to be incredibly reliable. Figure 3b illustrates the variation in the nanofluid velocity field due to the impact of Gr.In this scenario, the velocity field grows as the improved Gr . Physically, Gr decided how to account for buoyancy in terms of viscous force. As a result, as Gr changed, the velocity field enlarged, and the buoyancy force occurred. The effect of φ TiO 2 on nanofluids velocity distribution is shown in Fig. 3c. This graph analyzed the findings that the nanofluids speed drops as φ TiO 2 values rise. This visualizing suggests that increases in boundary layer surface thinning reflect fierce opposition to nanofluid speed. Figure 3d shows the visual findings for the velocity distribution under the variation of . It shown that when the value of pro- www.nature.com/scientificreports/ vides the maximum fluctuation, the velocity profile exhibits a decreasing behavior. Additionally, it is clear from Fig. 3a,b drawings that though the outputs with increases in φ TiO 2 are only very little different in both cases, they are significantly more explosive in the case of the C 2 H 6 O 2 base fluid when compared to H 2 O.
Impact of physical parameters on temperature distribution. Figure 4a-d are sketched for the nanofluids thermal distribution via Ec , Rd , φ TiO2 and S parameters. The effect of Ec on dimensionless temperature is clearly shown in Fig. 4a for both increasing and decreasing scenarios. It is revealed that nanofluid thermal profile has significant positive nature for raising values of Ec > 0 . The Eckert number, from a physical viewpoint, is a measure of the difference between the thermodynamic states of the fluid and the wall, which offers details about the fluid's self-heating properties in high-speed conditions. Frictional energy dissipated as the value of Ec rises because to the viscous interactions between fluid layers, heightening the nanofluid's temperatures. Considering such, the negative attitude is seen to be in contradiction with Ec < 0 . Figure 4b shows the Rd variability on the thermal profile of nanofluids. As Rd values grow, the temperature profile gets stronger. The influence of higher Rd values on conduction is dominant. The system receives a substantial amount of heat from the radiation, which raises the temperature. Figure 4c demonstrated that the result of φ TiO2 causes an increase in the temperature of nanofluids. The flow field produces thermal energy because of minuscule particles interactions, which raises the fluid's temperature. Hence the injection of φ TiO2 increases the transmission of energy. However, as seen in Fig. 4d, rising levels of S result in a decrease in thermal field. Additionally, like the fluid's velocity graphs, it is also noticeable from Fig. 4  Exploration of CVFEM findings. The illustrations in Fig. 5a-d explain the effects of fluid velocity and the magnetic field from partly and overall viewpoint. Figure 5a-d demonstrate the CVFEM views in portions and as a whole for the aforementioned features, separately. Two scenarios are taken into consideration to invigorate the results. In addition, Fig. 5a and c, which display the velocity contours, and Fig. 5b and d, which simultaneously display the magnetic profiles, respectively.

Conclusions
In the presence of thermal radiation, the simulation framework for the flow of H 2 O and C 2 H 6 O 2 -based TiO 2 nanofluids close to the stagnation point pattern with the Riga plate is examined. Analysis of the innovative effects of the current work is delighted with the unsteady relationships with dissipation. CVFEM and the RK-4 scheme are used to computationally solve the model differential equations. To examine the effects of innovative factors on various flow patterns, graphic findings and tabular data are presented. The key points are:   www.nature.com/scientificreports/ • The comparison of TiO 2 nanoparticles with H 2 O and C 2 H 6 O 2 base fluid is the only one used in the findings, which are, nevertheless, only compelling. If a different base fluid is utilized, the outcomes can be varied. The thermal development of diverse nanofluids must therefore be investigated in further research. Future studies may want to take the combination of water and ethylene glycol, magnetized hybrid nanofluid, and statistical and numerical data analysis into consideration.

Data availability
The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request.